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    The circle S1 with centre C1(a1,b1) radius r1, touches externally the circle S2 with centre C2(a2,b2) and radius r2. The tangent at their common point passes through the origin. Show that;

    (a1^2 - a2^2)+(b1^2-b2^2) = (r1^2-r2^2)

    Stuck... help would be appreciated
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    (Original post by Tiri)
    The circle S1 with centre C1(a1,b1) radius r1, touches externally the circle S2 with centre C2(a2,b2) and radius r2. The tangent at their common point passes through the origin. Show that;

    (a1^2 - a2^2)+(b1^2-b2^2) = (r1^2-r2^2)

    Stuck... help would be appreciated
    What have you tried and what level of maths do you have?

    This is easily solved using a diagram and a couple of applications of Pythagoras' Theorem.
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    Okay ill post pictures.. So far i know that the distance c1c2= r1+ r2,
    I initially thought that the distances formula subbed into this would give me the equation but it seems that the equations required looks like a difference of two squares of the radius's. This is a question from the 60s around a level stuff... Name:  ImageUploadedByStudent Room1420293973.581017.jpg
Views: 79
Size:  52.2 KB


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    Heres page two after i attempted to find the external touching point which i labeled P..Name:  ImageUploadedByStudent Room1420294020.285296.jpg
Views: 73
Size:  50.3 KB

    Then at this point i tried to sneakily find what the 2 radius's were and after squaring them tried subtracting them from eachother to give me the required equation but the calculations are nuts..Name:  ImageUploadedByStudent Room1420294109.901418.jpg
Views: 56
Size:  41.8 KB


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    (Original post by Tiri)
    Heres page two after i attempted to find the external touching point which i labeled P..Name:  ImageUploadedByStudent Room1420294020.285296.jpg
Views: 73
Size:  50.3 KB

    Then at this point i tried to sneakily find what the 2 radius's were and after squaring them tried subtracting them from eachother to give me the required equation but the calculations are nuts..Name:  ImageUploadedByStudent Room1420294109.901418.jpg
Views: 56
Size:  41.8 KB


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    Ok that all looks like massive overkill.

    I drew a pair of touching circles with a common tangent that passes through (0,0). I labelled the distance from (0,0) to the intersection point d. I then drew two lines from (0,0) to the centres of the circles. I then had a pair of back to back right angled triangles. I then applied Pythagoras to each triangle and subtracted the equations to eliminate d^2.
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    lol your a life saver... My head loves to drift away and overcomplicates things. Cheers!
 
 
 
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